MACRO DESIGN MODELS
FOR A SINGLE ROUTE
Outline
1. Introduction to analysis approach
2. Bus frequency model
3. Bus size model
4. Stop/station spacing model
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
1
Introduction to Analysis Approach
•
Basic approach is to establish an aggregate total cost function
including:
•
•
•
operator cost as f (design parameters)
user cost as g (design parameters)
Minimize total cost function to determine optimal design
parameter (s.t. constraints)
Variants include:
•
•
Maximize service quality s.t. budget constraint
Maximize consumer surplus s.t. budget constraint
Total Cost
Cost
User Cost
Operator Cost
Hop
T
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
Headway
2
Bus Frequency Model: the Square Root Model
Problem: define bus service frequency on a route as a
function of ridership
Total Cost = operator cost + user cost
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
3
Square Root Model (cont’d)
This is the Square Rule with the following implications:
•
high frequency is appropriate where
(cost of wait time/cost of operations time) is high
•
frequency is proportional to the square root of ridership per
unit time for routes of similar length
Frequency-Ridership Relationship
Frequency
Constant Load
Factor
Capacity
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
Ridership
4
Square Root Model (cont’d)
• Load factor is proportional to the square root of the
product of ridership and route length.
Bus Capacity-Ridership Relationship
Passengers/ bus
Load Factor >1
t1
t1>t2
Bus Capacity
t2
Ridership
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
5
Square Root Model (cont’d)
Critical Assumptions:
•
bus capacity is never binding
•
only frequency benefits are wait time savings
•
ridership ≠ f (frequency)
•
simple wait time model
•
budget constraint is not binding
Possible Remedies:
•
introduce bus capacity constraint
•
modify objective function
•
introduce r=f(h) and re-define objective function
•
modify objective function
•
introduce budget constraint
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
6
Bus Frequency Example
If:
c = $90/bus hour
b = $10/passenger hour
t = 90 mins
r = 1,000 passengers/hour
Then: hOPT = 11 mins
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
7
Bus Size Model
Problem: define optimal bus size on a route
Assumptions:
•
•
•
•
Desired load factor is constant
Labor cost/bus hour is independent of bus size
Non-labor costs are proportional to bus size
Bus dwell time costs per passenger are independent of bus size
Using same notation as before plus:
w = labor cost per bus hour
p = passenger flow past peak load point
k = desired bus load - the decision variable to be determined
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
8
Bus Size Model (cont’d)
Result is another square root model, implying that
optimal bus size increases with:
•
round trip time
•
ratio of labor cost to passenger wait time cost
•
peak passenger flow
•
concentration of passenger flows
Previous example extended with:
p
= 500 pass/hour
w
= $40/bus hour
all other parameters as before:
Then:hOPT
Nigel Wilson
= 55
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
9
Stop/Station Spacing Model
Problem: determine optimal stop or station spacing
Trade-off is between walk access time (which increases with station
spacing), and in-vehicle time (which decreases as station spacing
increases) for the user, and operating cost (which decreases as station
spacing increases)
Define
and
Nigel Wilson
Z
Tst
=
=
total cost per unit distance along route and per headway
time lost by vehicle making a stop
c
s
=
=
vehicle operating cost per unit time
station/stop spacing - the decision variable to be
determined
N
v
D
=
=
=
vacc
w
cs
=
=
=
number of passengers on board vehicle
value of passenger in-vehicle time
demand density in passenger per unit route length per
headway
value of passenger access time
walk speed
station/stop cost per headway
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
10
Stop/Station Spacing Model (cont’d)
Yet another square root relationship, implying that station/stop
spacing increases with:
•
•
•
•
•
•
walk speed
station/stop cost
time lost per stop
vehicle operating cost
number of passengers on board vehicle
value of in-vehicle time
and decreases with:
•
•
Nigel Wilson
demand density
value of access time
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
11
Bus Stop Spacing
U.S. Practice
•
•
•
200 m between stops (8 per mile)
shelters are rare
little or no schedule information
European Practice
•
•
•
•
320 m between stops (5 per mile)
named & sheltered
up to date schedule information
scheduled time for every stop
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
12
Stop Spacing Tradeoffs
Walking time
Riding time
Operating cost
Ride quality
Operator + User Cost
•
•
•
•
total extra cost
extra walk time
extra riding time
extra operating cost
Stop spacing (m)
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
13
Walk Access: Block-Level Modeling
a
Main Street
with Existing
Stops
Shed Line
b
Figure by MIT OpenCourseWare.
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
14
SCALE
0
0.5
MI
1
B BAY
NU
HEATH
10
8
6
4
2
F HILLS
S TO P S / M I
Results: MBTA Route 39*
Existing stop
Discrete model optimal stop
Existing route
Continuous model optimum
MBTA guideline
Discrete model optimum
Figure by MIT OpenCourseWare.
AM Peak Inbound results
•Avg walking time up 40 s
•Avg riding time down 110 s
•Running time down 4.2 min
•Save 1, maybe 2 buses
Source: Furth, P.G. and A. B. Rahbee, “Optimal Bus Stop Spacing Using Dynamic Programming and Geographic
Modeling." Transportation Research Record 1731, pp. 15-22, 2000.
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
15
Bus Stop Locations and Policies
•
Far-side (vs. Near-side)
•
•
•
•
less queue interference
easier pull-in
fewer ped conflicts
snowbank problem demands priority in maintenance
•
Curb extensions benefit transit, peds, and traffic
(0.9 min/mi speed increase)
•
Pull-out priority (it’s the law in some states)
•
Reducing dwell time (vehicle design, fare collection,
fare policy)
Nigel Wilson
1.258J/11.541J/ESD.226J
Spring 2010, Lecture 16
16
MIT OpenCourseWare
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1.258J / 11.541J / ESD.226J Public Transportation Systems
Spring 2010
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